To be precise, particles (with precise momentum) are certain modes or excitations of a quantum field which is often governed by something like a wave equation. However, not all quantum fields (and their associated particles) are force carriers. These are usually just the gauge bosons.
In any case, though, noone has managed to quantize gravity yet (i.e. describe gravity as a quantum field theory with gravitons as force carriers), so we've detected just gravitational waves but no gravitons.
Well, we've quantized gravitation extremely well - it's just perturbative quantum gravity, of which there are a couple flavours. In weak gravity it makes predictions identical to General Relativity, but is taken to be an effective field theory (in the Charles Wilson sense of "effective) in that we do not know how to deal with Feynman diagrams with more than a loop or two of gravitons, which only happens in strong gravity.
In these theories, gravitons are quantizations of the weak-field perturbations.
The non-renormalizability by power-set counting of gravity because it is a long-range force is a good result of perturbatively quantized gravity. Who knows if there is a workable way of renormalization by other methods? Not me.
Since we don't get strong gravity except close to a black hole singularity or in the extremely early universe, it's unfair and premature to say that the EFT approach has been unsuccessful.
Since QFT usually involves perturbation theory of a free field, whether or not particles are virtual (= intermediate products of interactions) is beside the point.
